On Jan 3, 4:52 pm, gus gassmann <g...@nospam.com> wrote: > On 03/01/2013 8:58 AM, Zuhair wrote: > > > > > > > > > > > On Jan 3, 3:23 pm, gus gassmann <g...@nospam.com> wrote: > >> On 03/01/2013 5:31 AM, Zuhair wrote: > > >>> Call it what may you, what is there is: > >>> (1) ALL reals are distinguishable on finite basis > > >> > (2) Distinguishability on finite basis is COUNTABLE. > > >> What does this mean? If you have two _different_ reals r1 and r2, then > >> you can establish this fact in finite time. The set of reals that are > >> describable by finite strings over a finite character set is countable. > >> However, not all reals have that property. > > > I already have written the definition of that in another post, and > > this post comes in continuation to that post, to reiterate: > > > r1 is distinguished from r2 on finite basis <-> > > Exist n. n is a natural number & Exist d_n, k_n: d_n is the n_th > > digit > > of r1 & k_n is the n_th digit of r2 & d_n =/= k_n) > > Exactly. This is precisely what I wrote. IF you have TWO *DIFFERENT* > reals r1 and r2, then you can establish this fact in finite time. > However, if you are given two different descriptions of the *SAME* real, > you will have problems. How do you find out that NOT exist n... in > finite time? > That is irrelevant (or at least that's how it appears to me) ANY two distinct (i.e. different) reals can be distinguished in finite time and any two reals that are distinguished in finite time are distinct. That's all what we need. The question of revealing the identity of some real in finite time is another matter, and my argument do not involve it at all, so it is irrelevant.
> Moreover, being able to distinguish two reals at a time has nothing at > all to do with the question of how many there are, or how to distinguish > more than two. Your (2) uses a _different_ concept of distinguishability.
(2) simple refers to how many finite initial segments of reals can be distinguishable? i.e. what is the total number of such segments. Clearly we have COUNTABLY many finite initial segments of reals. In other words we cannot distinguish more than COUNTABLY many finite initial segments of reals. Of course all of those are distinguished in finite time no doubt.
I think that distinguishability in (2) is the same distinguishability in (1) it has exactly the same definition.
We have only COUNTABLY many finite initial segments of reals that we can distinguish of course on finite basis, that's what is available, we don't have more.
Now every Two distinct reals are distinguishable on FINITE basis. But we have only Countably many finite initial segments of reals available for us to distinguish reals by, so how come we've ended up with UNCOUNTABLY many reals, what is the source of the excess in the number of reals, how can we distinguish more than what is available for us to distinguish. This is like saying that we have Three SEEDS, and Each Two distinct TREES grown from planting the three seeds must have Two distinct seeds where each Tree have grown from one seed, and then one comes and say that planting the three seeds had resulted in FOUR Trees? This is an example of a product outnumbering the potential of production?
By the way I might be wrong of course, I'll be glad to have anyone spot my error, my analogies might simply be misleading.